If $f$ is a locally integrable function then its Mellin transform
$\mathcal{M}[f]$ is defined by
$$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$
This integral usually converges in a strip $\alpha < Re \; s < \beta$ and
defines an analytic function. For our purposes we can assume that
$\mathcal{M}[f]$ converges in the right half-plane.

Now, let us assume that we are given a function $f (x)$ and its Mellin
transform $F (s)$. Further, suppose that we know that $F (s)$, just as the
gamma function, can be analytically extended to the whole complex plane with
poles at certain nonpositive integers. We also know that $F (s)$ satisfies a
functional equation which we would like to translate back into a differential
equation for $f (x)$. Formally, we obtain, say, a third order differential
equation with polynomial coefficients. Can we conclude that $f (x)$ solves
this DE?

The issue is that in our case the derivatives of $f (x)$ develop singularities
in the domain and are no longer integrable. So Mellin transforms can't be
defined in the usual way for them (and so we can't just use Mellin inversion).

What I am looking for is conditions under which we can still conclude that the
functional equation for $F (s)$ translates into a differential equation for $f
(x)$. Preferably, these should be conditions on $F (s)$ and not on $f (x)$. If
it helps, we can assume $f (x)$ to be compactly supported.

2 Answers
2

As long as the singularities are not "too bad", the answer will be yes, $f(x)$ will represent a solution of the differential equation. The very same way that
$$ \sum_{x=0}^{\infty} (-1)^{x}n!x^{n+1} $$
represents a solution of $x^2y'+y=x$. One might object that the series diverges, but resummation theory says this is irrelevant, that sum nevertheless represents a unique function (in a sector) which is a solution of the differential equation.

Balser's book, "From divergent series to analytic differential equations", would be a good place to start. Since the theory fundamentally uses Mellin transforms, you should be able to find what you want (perhaps indirectly) there.

Dear Jacques Carette -- I've tried following up on this and haven't been able to find anything related to Mellin transforms. Can you be more precise about where in his book they are discussed? (Also, which book? It seems you've combined the title of two of his books.) Thanks in advance.
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JBorgerAug 25 '10 at 6:50